Class-Numbers of Complex Quadratic Fields

Abstract

Let E be an elliptic curve

$$ y^2 = 4x^3 - g_2 x - g_3 ,\Delta = g_2^3 - 27g_3^2 \ne 0, $$

in Weierstrass normal form. The curve may be parametrized by the Weierstrass ℘-function, x = ℘(z), y = ℘′(z). The function ℘(z) is a doubly periodic function whose periods form a lattice

$$ \Lambda = \{ \omega _1 ,\omega _2 \} = \{ a\omega _1 + b\omega _2 |a,b \in \mathbb{Z}\} $$

where ω₁/ω₂ ∉ ℝ and for convenience we assume that ω₁ and ω₂ are so ordered that Im(ω₁/ω₂) > 0. We then have the relations

$$ g_2 = 60\sum\limits_\omega {'\omega ^{ - 4} } ,g_3 = 140\sum\limits_\omega {'\omega ^{ - 6} } , $$

(1)

where the summations are over all ω ∈ Λ other than ω = 0, and

$$ \wp (z) = \frac{1} {{z^2 }} + \frac{{g_2 }} {{20}}z^2 + \frac{{g_3 }} {{28}}z^4 + \frac{{g_2^2 }} {{1200}}z^6 + .... $$

(2)

.